Factorization and complex couplings in SYK and in Matrix Models

TL;DR

The work tackles the factorization problem in holographic toy models by recasting fixed-coupling theories as ensemble-averaged systems with a delta-function constraint, leading to a two-term decomposition of squared two-boundary observables: a wormhole contribution and a second term from a pair of linked half-wormholes. This second term can be interpreted as averaging over the imaginary parts of couplings, providing a unified perspective that applies to both SYK and matrix models, including GUE. The authors derive explicit constructions in SYK (with one time point and full theory) and in Gaussian matrix models, demonstrating that the leading large-$N$ behavior is captured by the proposed approximation and that the associated errors are suppressed. These results offer a potential bridge between fixed-coupling quantum systems and dynamical gravity, suggesting how factorization might be restored through fluctuations of otherwise fixed parameters and outlining several open directions for broader applicability and deeper gravity dual interpretations.

Abstract

We consider the factorization problem in toy models of holography, in SYK and in Matrix Models. In a theory with fixed couplings, we introduce a fictitious ensemble averaging by inserting a projector onto fixed couplings. We compute the squared partition function and find that at large $N$ for a typical choice of the fixed couplings it can be approximated by two terms: a "wormhole" plus a "pair of linked half-wormholes". This resolves the factorization problem. We find that the second, half-wormhole, term can be thought of as averaging over the imaginary part of the couplings. In SYK, this reproduces known results from a different perspective. In a matrix model with an arbitrary potential, we propose the form of the "pair of linked half-wormholes" contribution. In GUE, we check that errors are indeed small for a typical choice of the hamiltonian. Our computation relies on a result by Brezin and Zee for a correlator of resolvents in a "deterministic plus random" ensemble of matrices.